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\author{SSK }
\title{Python使用入门习题 }

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%\begin{abstract}
%Python编程练习
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%\section{}
\begin{enumerate}
\item  
在同一个图形界面，画出如下三个函数的图形，并进行标注：
$$y=\text{ch} x, y= \text{sh} x, y=e^x/2. $$

\item  
画出 $\Gamma$ 函数 $\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}dt$ 的图形。

\item  
在同一个图形界面中，分别画出6条曲线：
$$y=kx^2+2k, k=1,2,\cdots,6. $$

\item  
把屏幕开成2行3列6个子窗口，每个子窗口画一条曲线，画出曲线
$$y=kx^2+2k, k=1,2,\cdots,6. $$

\item  
分别画出下列二次曲面：
\begin{enumerate}
\item 单叶双曲面 $\frac{x^2}{4}+\frac{y^2}{10}-\frac{z^2}{8}=1$;
\item 椭圆抛物面 $\frac{x^2}{4}+\frac{y^2}{6}=z$.
\end{enumerate}

\item  
附件1：区域高程数据.xlsx给出了某区域的高程数据，画出该区域的三维表面图和等高线图，
在$A(30,0)$和$B(43,30)$（单位：公里）点处建立了两个基地，在等高线图上标注出这两个点。并求该区域地表面积的近似值。

\item  
先判断下列线性方程组的解的情况，然后求对应的唯一解、最小二乘解、或最小范数解。
\begin{eqnarray*}
(1)\left\{\begin{array}{rcl}
4x_1+2x_2-x_3 &=& 2, \\
3x_1-x_2+2x_3 &=& 10, \\
11x_1+3x_2 &=& 8. \\
\end{array}\right.
\hspace{1cm}
(2)\left\{\begin{array}{rcl}
2x+3y-z &=& 4, \\
x-2y+4z &=& -5, \\
3x+8y-2z &=& 13, \\
4x-y+9z &=& -6. \\
\end{array}\right.
\end{eqnarray*}

\item  
求解线性方程组
\begin{eqnarray*}
\left\{\begin{array}{rcl}
4x_1+x_2&=&1, \\ 
x_1+4x_2+x_3&=&2, \\ 
x_2+4x_3+x_4&=&3, \\ 
\cdots && \cdots \\
x_{998}+4x_{999}+x_{1000}&=&999, \\ 
x_{999}+4x_{1000}&=&1000. \\ 
\end{array}\right.
\end{eqnarray*}

\item  
求下列方程组的符号解和数值解
\begin{eqnarray*}
\left\{\begin{array}{rcl}
x^2-y-x&=&3, \\ 
x+3y&=&2.
\end{array}\right.
\end{eqnarray*}

\item  
某容器内侧是由曲线 $x^2+y^2=4y(1\le y\le 3)$ 与 $x^2+y^2=4(y\le 1)$ 绕$y$轴旋转一周而成的曲面。求容器的体积。若将容器内盛满的水从容器顶部全部抽出，至少需要做多少功？

\item  
已知 
\begin{eqnarray*}
f(x) &=& (|x+1|-|x-1|)/2 + \sin x, \\
g(x) &=& (|x+3|-|x-3|)/2 + \cos x, 
\end{eqnarray*}
求下列方程组的数值解。
\begin{eqnarray*}
\left\{\begin{array}{rcl}
2x_1 &=& 3f(y_1) +4g(y_2) -1, \\ 
3x_2 &=& 2f(y_1) +6g(y_2) -2, \\ 
y_1 &=& f(x_1) +3g(x_2) -3, \\ 
5y_2 &=& 4f(x_1) +6g(x_2) -1. \\ 
\end{array}\right.
\end{eqnarray*}

\item  
求下列矩阵的特征值和特征向量的数值解和符号解：
\begin{eqnarray*}
\begin{bmatrix}
-1&1&0 \\ 
-4&3&0 \\ 
1&0&2 \\ 
\end{bmatrix}.
\end{eqnarray*}

\item  
已知 
\begin{eqnarray*}
f(x) &=& (|x+1|-|x-1|)/2 + \sin x, \\
g(x) &=& (|x+3|-|x-3|)/2 + \cos x, 
\end{eqnarray*}
求下列超定方程组的最小二乘解。
\begin{eqnarray*}
\left\{\begin{array}{rcl}
2x_1 &=& 3f(y_1) +4g(y_2) -1, \\ 
3x_2 &=& 2f(y_1) +6g(y_2) -2, \\ 
y_1 &=& f(x_1) +3g(x_2) -3, \\ 
5y_2 &=& 4f(x_1) +6g(x_2) -1. \\ 
x_1+y_1 &=& f(y_2) +g(x_2) -2, \\ 
x_2-3y_2 &=& 2f(x_1) -10g(y_1) -5. \\ 
\end{array}\right.
\end{eqnarray*}


\end{enumerate}

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\end{document}

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